\left\{ \begin{array} { c } { x + y = 90 } \\ { 900 x + 1200 y = 55 } \end{array} \right.
Solve for x, y
x = \frac{21589}{60} = 359\frac{49}{60} \approx 359.816666667
y = -\frac{16189}{60} = -269\frac{49}{60} \approx -269.816666667
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x+y=90,900x+1200y=55
To solve a pair of equations using substitution, first solve one of the equations for one of the variables. Then substitute the result for that variable in the other equation.
x+y=90
Choose one of the equations and solve it for x by isolating x on the left hand side of the equal sign.
x=-y+90
Subtract y from both sides of the equation.
900\left(-y+90\right)+1200y=55
Substitute -y+90 for x in the other equation, 900x+1200y=55.
-900y+81000+1200y=55
Multiply 900 times -y+90.
300y+81000=55
Add -900y to 1200y.
300y=-80945
Subtract 81000 from both sides of the equation.
y=-\frac{16189}{60}
Divide both sides by 300.
x=-\left(-\frac{16189}{60}\right)+90
Substitute -\frac{16189}{60} for y in x=-y+90. Because the resulting equation contains only one variable, you can solve for x directly.
x=\frac{16189}{60}+90
Multiply -1 times -\frac{16189}{60}.
x=\frac{21589}{60}
Add 90 to \frac{16189}{60}.
x=\frac{21589}{60},y=-\frac{16189}{60}
The system is now solved.
x+y=90,900x+1200y=55
Put the equations in standard form and then use matrices to solve the system of equations.
\left(\begin{matrix}1&1\\900&1200\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}90\\55\end{matrix}\right)
Write the equations in matrix form.
inverse(\left(\begin{matrix}1&1\\900&1200\end{matrix}\right))\left(\begin{matrix}1&1\\900&1200\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\900&1200\end{matrix}\right))\left(\begin{matrix}90\\55\end{matrix}\right)
Left multiply the equation by the inverse matrix of \left(\begin{matrix}1&1\\900&1200\end{matrix}\right).
\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\900&1200\end{matrix}\right))\left(\begin{matrix}90\\55\end{matrix}\right)
The product of a matrix and its inverse is the identity matrix.
\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}1&1\\900&1200\end{matrix}\right))\left(\begin{matrix}90\\55\end{matrix}\right)
Multiply the matrices on the left hand side of the equal sign.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{1200}{1200-900}&-\frac{1}{1200-900}\\-\frac{900}{1200-900}&\frac{1}{1200-900}\end{matrix}\right)\left(\begin{matrix}90\\55\end{matrix}\right)
For the 2\times 2 matrix \left(\begin{matrix}a&b\\c&d\end{matrix}\right), the inverse matrix is \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right), so the matrix equation can be rewritten as a matrix multiplication problem.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4&-\frac{1}{300}\\-3&\frac{1}{300}\end{matrix}\right)\left(\begin{matrix}90\\55\end{matrix}\right)
Do the arithmetic.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\times 90-\frac{1}{300}\times 55\\-3\times 90+\frac{1}{300}\times 55\end{matrix}\right)
Multiply the matrices.
\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{21589}{60}\\-\frac{16189}{60}\end{matrix}\right)
Do the arithmetic.
x=\frac{21589}{60},y=-\frac{16189}{60}
Extract the matrix elements x and y.
x+y=90,900x+1200y=55
In order to solve by elimination, coefficients of one of the variables must be the same in both equations so that the variable will cancel out when one equation is subtracted from the other.
900x+900y=900\times 90,900x+1200y=55
To make x and 900x equal, multiply all terms on each side of the first equation by 900 and all terms on each side of the second by 1.
900x+900y=81000,900x+1200y=55
Simplify.
900x-900x+900y-1200y=81000-55
Subtract 900x+1200y=55 from 900x+900y=81000 by subtracting like terms on each side of the equal sign.
900y-1200y=81000-55
Add 900x to -900x. Terms 900x and -900x cancel out, leaving an equation with only one variable that can be solved.
-300y=81000-55
Add 900y to -1200y.
-300y=80945
Add 81000 to -55.
y=-\frac{16189}{60}
Divide both sides by -300.
900x+1200\left(-\frac{16189}{60}\right)=55
Substitute -\frac{16189}{60} for y in 900x+1200y=55. Because the resulting equation contains only one variable, you can solve for x directly.
900x-323780=55
Multiply 1200 times -\frac{16189}{60}.
900x=323835
Add 323780 to both sides of the equation.
x=\frac{21589}{60}
Divide both sides by 900.
x=\frac{21589}{60},y=-\frac{16189}{60}
The system is now solved.
Examples
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{ x } ^ { 2 } - 4 x - 5 = 0
Trigonometry
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Linear equation
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Arithmetic
699 * 533
Matrix
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Simultaneous equation
\left. \begin{cases} { 8x+2y = 46 } \\ { 7x+3y = 47 } \end{cases} \right.
Differentiation
\frac { d } { d x } \frac { ( 3 x ^ { 2 } - 2 ) } { ( x - 5 ) }
Integration
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Limits
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